Thermodynamics and galactic clustering with a modified gravitational potential Sudhaker Upadhyay∗
arXiv:1703.06781v1 [gr-qc] 17 Mar 2017
Centre for Theoretical Studies, Indian Institute of Technology Kharagpur, Kharagpur-721302, India Based on thermodynamics, we study the galactic clustering of an expanding Universe by considering the logarithmic and volume (quantum) corrections to Newton’s law along with the repulsive effect of a harmonic force induced by the cosmological constant (Λ) in the formation of the large scale structure of the Universe. We derive the N -body partition function for extended-mass galaxies (galaxies with halos) analytically. For this partition function, we compute the exact equations of states, which exhibit the logarithmic, volume and cosmological constant corrections. In this setting, a modified correlation (clustering) parameter (due to these corrections) emerges naturally from the exact equations of state. We compute a corrected grand canonical distribution function for this system. Furthermore, we obtain a deviation in differential forms of the two-point correlation functions for both the point-mass and extended-mass cases. The consequences of these deviations on the correlation function’s power law are also discussed.
Keywords: Cosmology; Modified gravity; Galaxies cluster; Large scale structure of universe; Correlation function; Distribution function.
I.
OVERVIEW AND MOTIVATION
The characterization of galactic clusters on very large scales under the influence of their mutual gravitational interaction is a matter of vast interest. The importance of such a process can be exaggerated as the evolution and distribution of the galaxies throughout the Universe are the main manifestations of this. The analysis of the correlation functions is one of the standard ways to study the formation of the Universe. The observation tells us that the power law of two-point correlation function scales as (intergalactic distance)−1.6 to (intergalactic distance)−1.8 [1], which has also been approved by N -body computer simulations [2] and by the analytic gravitational quasiequilibrium thermodynamics [3]. The calculation of the power law of the correlation function is based on the assumption that the conversion of the initial primordial matter into the observed many-body galaxies took place at the stage of evolution of the Universe and these galaxies are coupled to the expansion of the Universe. The theories of the many-body (galaxies) distribution function have been developed mainly from a thermodynamic point of view [3–8]. Theses theories utilize only the first two laws of thermodynamics to derive the exact equations of state of the expanding Universe (a quasiequilibrium evolution). The relation between thermodynamics and relativity was originated in the work of Bekenstein [9], Hawking [10], and Unruh [11]. Later, Jacobson established an important connection between thermodynamics and general relativity, by which the Einstein equations themselves can be viewed as a thermodynamic equation of state under a set of minimal assumptions involving the equivalence principle and the identification of the area of a causal horizon with entropy [12–16]. Recently, Verlinde proposed a constructive idea stating that gravity is not a fundamental interaction and can be interpreted as an entropic force [17]. Although this idea fails to provide a rigorous physical explanation [18, 19], it surely opens a new window into understanding gravity from first principles. For instance, the modified Newton’s law [20], Friedmann equations at the apparent horizon of the Friedmann-Robertson-Walker Universe [21, 22], modified Friedmann equations [23, 24], Newtonian gravity in loop quantum gravity [25], holographic dark energy [26–28], thermodynamics of black holes [29], and extension to Coulomb force [30], etc. support the entropic interpretation of gravity.
∗ Electronic
address:
[email protected]
2 Verlinde’s approach to get Newton’s law of gravity relies on the entropy-area relation of black holes in Einstein’s gravity, i.e., S = 4lA2 , where S is the entropy of the black hole, A is the area of the horizon, and P lP is the Planck length. In order to include the quantum corrections to the area law, some modifications are required to the area law [31]. In the literature, the two well-studied quantum corrections to the area law are, namely, the logarithmic correction and power law correction. The logarithmic correction appears due to the thermal equilibrium fluctuations and the quantum fluctuations of loop quantum gravity [32– 34]. However, the power law correction appears due to the entanglement of quantum fields sitting near the horizon [35, 36]. Recently, combined corrections due to the logarithmic and volume terms are proposed for Newton’s law of gravitation as [20] 3 A A 2 A S = 2 − a log +b , (1) 4lP 2lP2 2lP2 where a and b are the constants of the order of unity or less. Here, the second term of the rhs corresponds to the logarithmic correction and the last term of the rhs corresponds to the volume correction. Modesto and Randono [20] discussed how deviations from Newton’s law caused by the logarithmic correction have the same form as the lowest-order quantum effects of perturbative quantum gravity; however, the deviations caused by the volume correction follow the form of the modified Newtonian gravity models explaining the anomalous galactic rotation curves. In fact, on the very large (cosmological) scale, it is expected or otherwise speculated that the cosmological constant, as a prime candidate for dark energy, is responsible for the expansion of the Universe through a repulsive force [37]. If the effect of the cosmological constant in the Newtonian limit of a metric is present in some phenomenon (such as clustering of galaxies), then the same effect should also be present in the full general relativistic treatment of the same phenomenon. Keeping the importance of the cosmological constant in mind, we want to employ the cosmologicalconstant-induced (harmonic-oscillator-type) modification to Newton’s law, which leads to the cosmic repulsive force. Our motivation here is to study the effects of such corrections on the characterization of the clustering of galaxies on very large scales. The clustering of galaxies within the framework of modified Newton’s gravity through the cosmological constant only has been studied very recently[38]. To study the effects of the logarithmic, volume, and the cosmological constant corrections to the clustering of galaxies, we first derive the N -body partition function by evaluating configuration integrals recursively. From the resulting partition function, we extract various thermodynamical (exact) equations of state. For instance, we compute the Helmholtz free energy, entropy, pressure, internal energy, and chemical potential, which possess deviations from their original values due to the logarithmic, volume, and cosmological constant corrections. Remarkably, a modified correlation (clustering) parameter emerges naturally from the more exact equations of state. In the limit a → 0, b → 0, and Λ → 0, the modified correlation parameter coincides with its original value given in [5]. By assuming that the system is in a quasiequilibrium state as described by the grand canonical ensemble, we derive the probability distribution function. The resulting distribution function depends on the modified clustering parameter. Comparative analyses are made to see the effect of corrections on the probability distribution function. In this regard, we find that the corrected distribution function first increases sharply with the number of particles (N ) and gets a maximum (peak) value for the particular N . As long as N increases further beyond that particular value, the value of the distribution function starts descending very fast and becomes slow later. The peak value of the corrected distribution function decreases gradually with the increasing values of logarithmic and volume corrections. Remarkably, the highly corrected distribution function starts dominating the lesser corrected distribution function after a certain value of N . Due to the cosmological constant term, the peak value of the corrected distribution function decreases even further and falls rather slowly. We compute the differential form of the two-point correlation function for both the cases of point-mass and extended-mass galaxies. By solving the corresponding differential equation, we obtain a modified structure of the two-point correlation function for both the point-mass and the extended-mass galaxies. It is shown that the corrections also affect the power law of the correlation function. Although there are corrections to the power law behavior, the correlation function obeys the original result (as in Refs. [1–3]) under certain approximations. The paper is organized as follows. In Sec. II, we derive the N -body partition function for the gravitationally interacting system with the corrected Newtonian dynamics using the logarithmic, volume,
3 and cosmological constant terms. The thermodynamical properties and distribution functions for such a system are discussed in Sec. III. The differential form of the two-point correlation functions for both the point-mass and extended-mass galaxies are computed in Sec. IV. Within this section, the effect of corrections on the power law behavior of the two-point correlation functions is also discussed. Finally, the discussions and conclusions are made in the last section.
II.
INTERACTION OF GALAXIES THROUGH MODIFIED POTENTIAL
In this section, we consider a modified Newton’s law of gravitation due to the first-order corrections and study their effects on the partition function.
A.
A modified Newton’s law of gravitation
It has been stressed [39] that different quantum theories of gravity may lead to different higher-order corrections to the area law of Bekenstein-Hawking entropy. These corrections may display differences and, more interestingly, relations among quantizations. In [39], Kaul and Majumdar computed the lowestorder corrections to the area law in a particular formulation [40] of a quantum geometry program. They found that the leading correction is logarithmic, with ∆S ∼ log(A/2lP2 ). On the other hand, in loop quantum gravity, the entropy introduces a dependence on the number of loops L for the spin-network state dual to a region of surface [41]. In the limit of a larger number of loops L >> n, where n is the number of boundary edges, the entropy behaves as S(L >> n) ∼ n log L ∼ n3/2 ∝ A3/2 , where L has √ n exponential growth of the type L ∼ 2 . With these types of leading-order corrections to entropy, the expression of the (modified) area law results in (1). 2
∂S In fact, the Newtonian force (F) in terms of entropy reads F = −4lP2 GM R2 ∂A . Therefore, corresponding to the logarithmic and the volume corrected entropy (1), Newton’s force law gives √ R l2 GM 2 . (2) F = − 2 1 − a P 2 + b12 π R πR lP R This leads to the following corrected gravitational potential energy (Φ = − FdR) [20]: √ 1 12 π l2 R Φ = −GM 2 , (3) log −a P 3 −b R 3πR lP l
where l is an integration constant which signifies to (an unspecified) length parameter. However, at the cosmological scale, it is speculated that the cosmological constant Λ is responsible for the expansion of the Universe through a repulsive force. For example, the Schwarzschild–de Sitter spacetime in its static form is given by the following line element [42]: ds2 = f (R)dt2 −
1 dR2 − R2 (dθ2 + sin2 θdφ2 ), f (R) 2GM ΛR2 . f (R) = 1 − − R 3
Here we considered velocity of light c = 1. From the line element, it is natural to include an extra Λ-induced harmonic-oscillator-type potential − 61 ΛR [37] to (3). Therefore, by incorporating the cosmological-constant-induced modification to Newton’s law, the potential energy finally reads √ 1 ΛR2 12 π l2 R 1 + . (4) log −a P 3 −b Φ = −GM 2 R 3πR lP l 6 GM 2 In the next subsection, we will see the effect of these modifications on the many-body partition function.
4 B.
The partition function
The statistical mechanics of an N -body system is primarily based on the partition function. Here we note that all our analyses are based on the assumption that our gravitational system has a statistically homogeneous distribution over large regions, which consists of an ensemble of cells having the same volume V and the same average density ρ¯(N/V ). In order to deal with the galactic clustering from the statistical mechanics perspective, we first need to know the partition function (ZN (T, V )) of the gravitationally interacting system, which consists of N particles of equal mass M , momenta pi and average temperature T . This is generally given by [5] Z 1 H ZN (T, V ) = 3N d3N p d3N R exp − , (5) λ N! T where N ! corresponds to the distinguishability of classical particles and λ is a normalization constant which results space. Here the N -body Hamiltonian has the following from the integration over momentum PN p2i form: H = i=1 2M +Φ(r1 , r2 , ..., rN ) . In general, the gravitational potential energy, Φ(r1 , r2 , . . . , rN ), depends on the relative position vector of the ith and jth particles (i.e., R = |ri −rj |) and, hence, describes the sum of the potential energies of all pairs. Therefore, Φ(r1 , r2 , . . . , rN ) can be expressed as X X Φ(r1 , r2 , . . . , rN ) = Φij (R) = −T log(1 + fij ). (6) 1≤i
